ECAT cheat sheet

Newton's second law: net force equals mass times acceleration (F = ma); weight is mg, distinct from mass.

Kinematic equations apply only for constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as.

Projectile motion: horizontal velocity stays constant; vertical motion is free fall with a = g downward, so time of flight depends only on vertical component.

Conservation of momentum: total momentum of an isolated system is unchanged in collisions; kinetic energy is conserved only in elastic collisions.

Ohm's law: V = IR; resistors in series add directly, while resistors in parallel add reciprocally (1/R = 1/R₁ + 1/R₂).

Work-energy theorem: net work done on an object equals its change in kinetic energy (W = ΔKE = ½mv² − ½mu²).

For a simple pendulum, period T = 2π√(L/g), independent of mass and (for small angles) of amplitude.

Snell's law: n₁sinθ₁ = n₂sinθ₂; total internal reflection occurs only when light travels from denser to rarer medium beyond the critical angle.

Mole concept: one mole contains 6.022 × 10²³ particles; moles = mass / molar mass.

Avogadro's law and molar volume: at STP, one mole of any ideal gas occupies about 22.4 L.

pH = −log[H⁺]; pH below 7 is acidic, above 7 is basic, and pH + pOH = 14 at 25 °C.

Le Chatelier's principle: a system at equilibrium shifts to oppose a change in concentration, pressure, or temperature.

Oxidation is loss of electrons (increase in oxidation number); reduction is gain of electrons (decrease in oxidation number).

Ideal gas law: PV = nRT, with R ≈ 8.314 J/(mol·K); temperature must be in kelvin.

Periodic trends: atomic radius decreases across a period and increases down a group; electronegativity and ionization energy generally increase across a period.

Balancing redox equations requires equal electron loss and gain; in acidic solution, balance O with H₂O and H with H⁺.

Quadratic formula: for ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] / (2a); the discriminant b² − 4ac determines the nature of roots.

Sum and product of roots: for ax² + bx + c = 0, sum = −b/a and product = c/a.

Fundamental identity: sin²θ + cos²θ = 1; also 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.

Derivative rules: d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x.

Product rule: (uv)′ = u′v + uv′; quotient rule: (u/v)′ = (u′v − uv′)/v².

Arithmetic series sum: Sₙ = (n/2)[2a + (n−1)d]; geometric series sum: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.

Logarithm rules: log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n log x.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)² + (y₂−y₁)²]; slope m = (y₂−y₁)/(x₂−x₁).

A complete sentence needs a subject and a finite verb and must express a complete thought; a fragment lacks one of these.

Subject-verb agreement: a singular subject takes a singular verb; intervening phrases do not change the number of the subject.

Active voice: subject performs the action; passive voice: object becomes subject and the verb uses a form of 'be' plus past participle.

Direct to indirect speech shifts tense back one step and adjusts pronouns and time words (e.g., 'today' becomes 'that day').

Comma splices join two independent clauses with only a comma; fix with a period, semicolon, or coordinating conjunction.

Restrictive clauses (essential information) take no commas; non-restrictive clauses are set off by commas.

Commonly confused pairs: 'affect' is usually a verb, 'effect' usually a noun; 'its' is possessive, 'it's' means 'it is'.

Parallel structure requires items in a list or comparison to share the same grammatical form.